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For every natural number n, which of the following inequalities is true?

1. 2ⁿ < n
2. n² > 2n
3. n⁴ < 10ⁿ
4. 2³n > 7n + 1

Correct answer: n⁴ < 10ⁿ

Solution

2^n<n fails at n=1; n^2>2n fails at n=1,2; 8^n>7n+1 fails at n=1 (8=8). Only n^4<10^n holds for every natural n (1<10, 16<100, 81<1000, and 10^n grows far faster thereafter).

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